What is the order of operations?

Because mathematical expressions involving multiple operations could otherwise be evaluated to different values by different reasonable people, we have an agreement on the specific order in which operations are to be performed. This assures we will all get the same result when evaluating the same expression, assuming we are using the same agreement.

Although alternative agreements do exist (some programming languages and software packages are examples,) it is by and large the following agreement that is most commonly used in a classroom setting and thus the usual convention on alt.algebra.help:

1. Simplify inside grouping symbols, working from the innermost to the outermost. Grouping symbols include parentheses, brackets, and the fraction bar.

2. Simplify powers/roots.

3. Perform multiplication/division as they occur from left to right.

4. Perform addition/subtraction as they occur from left to right.

Examples:

$$6 \div 3+2^3 \cdot 5 = 6 \div 3+ \mathbf{8} \cdot 5$$

$$= \mathbf{2}+8 \cdot 5$$

$$= 2+ \mathbf{40}$$

$$= 42$$

$$(8+6) \div 7 \cdot 3- 6 = \mathbf{14} \div 7 \cdot 3 - 6$$

$$= \mathbf{2} \cdot 3 - 6$$

$$= \mathbf{6} - 6$$

$$= 0$$

$$\frac{-(-3)^3+(-5)}{2(-8)-5(3)} = \frac{-(\mathbf{-27})+(-5)}{2(-8)-5(3)}$$

$$= \frac{\mathbf{27}-\mathbf{5}}{\mathbf{-16-15}}$$

$$= \frac{22}{-31}$$

$$= -\frac{22}{31}$$